As is known, the keys of a piano are divided into seven groups or octaves. Each octave has twelve keys that are tuned with equal temperament. Equal temperament tuning is a method of tuning each of the notes within each octave such that every pair of adjacent notes has an identical frequency ratio. Pitch is perceived roughly as the logarithm of frequency by the human ear and thus the perceived “distance” from every note to its nearest neighbor is equal to a ratio of the logarithms of frequencies of the adjacent keys. Each frequency for each key of a piano is determined by the equation:
      f    ⁡          (      N      )        =            f      ⁡              (        M        )              *          (              2                  (                                    N              -              M                        12                    )                    )                      Where:        f(N) is the frequency of the Nth key of the piano keyboard.        N is the numbering of the keys of the piano with the farthest left key being N=0.        f(M) is the frequency of a key tuned to a standard pitch.        M is the key number of the key tuned to the standard pitch. Usually the standard pitch or frequency is the A key above middle C or A4. M=49 for the key A4.        
The history of the evolution of equal temperament is divided into two parts, before and after invention of equal temperament formula. First recorded date of the development of the concept of equal temperament is 1584 by Chu Tsai-Yu of the Ming Dynasty. Then European mathematicians Simon Stevin (1585) and Marin Mersenne (1636) developed their versions of equal temperament. Before equal temperament, it was a struggle to find some kind of universal system for tuning a musical instrument After introduction of the equal temperament concept, it was about 300 years before it was finally accepted for use in a few selected countries. It was assumed that J. S. Bach had intended equal temperament in his “Well-Temperd Clavier”. However, there is a difference between “well tempered” and “equal temperament”. The evolution of equal temperament still took decades for it to be permanently adopted by the musicians. In 1939, the standard pitch or frequency of the A key above middle C or A4 as 440 Hz was adopted.
FIG. 1 is a diagram of a piano keyboard showing the designations for each of the keys of the keyboard. To tune a piano minute adjustments are made to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is in tune. In the context of piano tuning, being in tune is not simply setting the tensions of the strings of the piano to a set of assigned frequencies or pitches. A piano tuner must assess the interaction between the notes. The tuner will tune one of the keys to a standard pitch or frequency such as the “A” key above middle “C” or “A4” that is tuned to a frequency of 440 Hz. The remaining keys are tuned in relationship to the chosen fixed pitch (i.e. “A4”=440 Hz).